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BMC: Distribution functions

So, for today’s BMC post, I’m going to be attacking a topic from a different angle. I’d like to show you all about thermodynamic distribution functions (mathematical functions that describe the number of particles across a range of some property). However, rather than simply explaining the properties to you in text, I’d like to tell you what the distributions describe, and then see what kind of properties we can derive from looking at the shape of the functions.

Boltzmann Distribution

The Boltzmann distribution is used to describe the arrangement of particles that are totally distinguishable (no identical particles), and that can fit in any particular energy level.

Boltzmann Distribution for different temperatures

Boltzmann Distribution for different chemical potentials

As the plots show, the Boltzmann distribution absolutely explodes as you approach the chemical potential (from the right). This tells you that the majority of the particles will stick around at low energy levels, unless the temperature is high. At high temperatures, the distribution smooths out: in other words, high temperature particles are more evenly distributed across energy levels. The chemical potential clearly shifts the graph along the Energy axis without changing its shape: this is an intuitive way of thinking of chemical potential: a constant push in one direction or another due to the interaction of the particles with the rest of the system.

Bose Distribution

The Bose distribution is used as a description of indistinguishable particles that do not interact with each other (they can have an arbitrary number per energy state).

Bose Distribution for different temperatures

Bose Distribution for different chemical potentials

As these plots show, the Bose distribution is almost identical to the Boltzmann distribution, but with one major difference: the distribution does not extend lower than the chemical potential. This is an interesting property, one we wouldn’t have expected just from the plain-language definition of the Bose distribution. This means that the Bose distribution excludes particles from energy states lower than the chemical potential. Bizarre!

Fermi Distribution

The Fermi distribution extends the Bose distribution with an additional attribute: the identical particles can only fit one at a time into a given energy state.

Fermi Distribution for different temperatures

Fermi Distribution for different chemical potentials

So, as you can see, these plots are quite different than the previous two. Rather than blowing up as the energy drops towards zero, it instead levels out to 1. This is because of the exclusionary property of fermions: they can’t occupy the same spot, so you can never find more than 1 in a given state. As you can see, at lower temperatures, the distribution looks like a step function, with 1 particle “locked” in below the chemical potential. At higher temperatures, more particles spread out into different energy levels (as we would expect). As the second plot shows, the behavior is the same as the other plots: the chemical potential simply determines at what level most of the particles get locked into.

So there you have it: you now know a little bit about each of these three different statistical distributions, and all from just looking at what they look like when graphed. Just looking at the shape of a function can tell you a lot about it.

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About bwkeller

I am an astrophysics graduate student at McMaster University. I got my BSc in physics with a computer science minor from the University of Calgary. I consider myself a skeptic, maker, and all-around nerd.